In this paper, we study generalized pseudostandard words over a
two-letter alphabet, which extend the classes of standard Sturmian,
standard episturmian and pseudostandard words, allowing different
involutory antimorphisms instead of the usual palindromic closure or a
fixed involutory antimorphism. We first discuss
pseudoperiods, a useful tool for describing words obtained by
iterated pseudopalindromic closure. Then, we introduce the concept of
normalized directive bi-sequence (Θ, w) of a generalized
pseudostandard word, that is the one that exactly describes all the
pseudopalindromic prefixes of it. We show that a directive bi-sequence
is normalized if and only if its set of factors does not intersect a
finite set of forbidden ones. Moreover, we provide a construction to
normalize any directive bi-sequence. Next, we present an explicit
formula, generalizing the one for the standard episturmian words
introduced by Justin, that computes recursively the next prefix of a
generalized pseudostandard word in term of the previous one. Finally,
we focus on generalized pseudostandard words having complexity 2n,
also called Rote words. More precisely, we prove that the
normalized bi-sequences describing Rote words are completely
characterized by their factors of length 2.